Are there any interesting non-Abelian examples of groups which are equal to the union of a dense chain of normal subgroups, with each subgroup isomorphic to the original group. That is, a group $G$ and set of normal subgroups $\{ H_{i}: i \in \mathbb{Q} \} $ such that:
1) $G=\bigcup_{i\in \mathbb{Q}} H_i$;
2) $H_i \subset H_j$ if and only if $i<j$];
3) For each $i$ we have $H_i\cong G$.
4) $G/H_i\cong G$.
One (although Abelian) example is $\bigoplus_{i\in \mathbb{Q}} \mathbb{Z}_2$; however this is isomorphic to $\bigoplus_{i\in \mathbb{N}} \mathbb{Z}_2$ (I believe), which wouldn't work...
The problem arises from a series of surjective (group) homomorphisms $ \psi_{i}:G\rightarrow G_i$ ($i\in \mathbb{Q}$) of a group $G$ such that 1) -4) hold for normal subgroups $\{\text{Ker } \psi_i:i\in \mathbb{Q}\}$.
Note: $G$ will clearly not be hopfian or co-hopfian.